Form Of Infinite Series Research Articles

Mathematical Theory of Transversally Isotropic Shells of Arbitrary Thickness at Static Load

Classical and non-classical refined theories of plates and shells, based on various hypotheses [1-7], for a wide class of boundary problems, can not describe with sufficient accuracy the SSS of plates and shells. These are boundary problems in which the plates and shells undergo local and burst loads, have openings, sharp changes in mechanical and geometric parameters (MGP). The problem also applies to such elements of constructions that have a considerable thickness or large gradient of SSS variations. The above theories in such cases yield results that can differ significantly from those obtained in a three-dimensional formulation. According to the logic in such theories, the accuracy of solving boundary problems is limited by accepted hypotheses and it is impossible to improve the accuracy in principle. SSS components are usually depicted in the form of a small number of members. The systems of differential equations (DE) obtained here have basically a low order. On the other hand, the solution of boundary value problems for non-thin elastic plates and shells in a three-dimensional formulation [8] is associated with great mathematical difficulties. Only in limited cases, the three-dimensional problem of the theory of elasticity for plates and shells provides an opportunity to find an analytical solution. The complexity of the solution in the exact three-dimensional formulation is greatly enhanced if complex boundary conditions or physically nonlinear problems are considered. Theories in which hypotheses are not used, and SSS components are depicted in the form of infinite series in transverse coordinates, will be called mathematical. The approximation of the SSS component can be adopted in the form of various lines [9-16], and the construction of a three-dimensional problem to two-dimensional can be accomplished by various methods: projective [9, 14, 16], variational [12, 13, 15, 17]. The effectiveness and accuracy of one or another variant of mathematical theory (MT) depends on the complex methodology for obtaining the basic equations.

Evaluation of thermoelastic damping in micromechanical resonators with axial pretension: An analytical model accounting for two-dimensional thermal conduction

It has been reported that application of tensile axial stress can simultaneously increase quality factor and resonant frequency for micromechanical resonators. In this study, we formulate an analytical model for evaluating thermoelastic damping in micromechanical resonators based on the thermal energy method, in which thermal conductions in both thickness direction and axial direction are considered. An explicit expression for thermoelastic damping in the form of infinite series has been obtained. The proposed analytical model is further validated by finite element analysis. Results of the present study demonstrate that the 2D model needs to be adopted in order to accurately evaluate thermoelastic damping of micromechanical resonators with axial pretension. In addition, the 2D model proposed in the present study eliminates the inherent inconsistency entailed in the 1D model.

Theory of Scalar Wave Scattering by a Planar Substrate Containing a Sphere

The problem of scalar wave scattering by a planar substrate containing a sphere is analytically solved. The wave functions, as the solution, come in the form of infinite series. The factors in each term of the series explicitly show the involved process and the multiplicity of scattering between the surface of the substrate and the sphere. In air, the incident wave is either reflected by or transmitted into the substrate. The transmitted wave is either scattered by or transmitted into the sphere, and then the backwardly scattered wave going to the surface of the substrate is either transmitted back into air or reflected toward the sphere. The reflected wave is again either scattered by or transmitted into the sphere, and so on. Within the substrate, the scattering by the sphere and the reflection at the planar surface repeat in turn indefinitely to generate multiply scattered waves, which combine to make different wave functions for different regions in the form of infinite series.

MGF‐based analysis of κ‐μ/gamma composite fading model for indoor off body communication

AbstractThis paper presents the moment generating function (MGF)–based analysis of κ‐μ/gamma composite fading model, which is suitable for the indoor off‐body communication channel. In this paper, first of all, we derive an analytical expression for MGF of instantaneous signal‐to‐noise ratio. By using the derived MGF, the expressions of average symbol error rate (ASER)/average bit error rate (ABER) for binary and M‐ary modulation scheme are derived. Using the MGF approach, we also derive the expression of channel capacity for optimum rate adaption and channel inversion with fixed rate policy. The derived expressions for ABER/ASER and channel capacity are presented in the infinite series form. We further present the numerical results of error rate and channel capacity for different values of κ, μ, α, and β parameters. The result of ABER and channel capacity shows good agreement with the previously reported results as special cases that verify the derived expressions.

Thermoelastic Instability in the Quasi-Static Coupled Thermoelasticity Problem Dealt with the Sliding Contact with Frictional Heating

A quasi-static coupled contact problem of thermoelasticity that deals with a sliding frictional contact with taking into account the frictional heating is considered. Exact solutions of the problem are constructed in the form of Laplace convolutions, after calculating which the solution has been written in form of infinite series over eigenvalues of problem. The study of these eigenvalues in relation to three dimensionless parameters of the problem is carried out. Based on the analysis of the solutions obtained, it is possible to distinguish the domains of stable and unstable solutions in the space of dimensionless parameters. The properties of the obtained solutions are studied in relation to the dimensional and dimensionless parameters of the problem. Within the framework of the main research problem, partial problems of monitoring the sliding parameters as well as problems of controlling contact parameters in order to avoid thermoelastic instability are formulated and solved.

Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection heat transfer

AbstractPhysiological transport phenomena often feature ciliated internal walls. Heat, momentum, and multispecies mass transfer may arise and additionally non‐Newtonian biofluid characteristics are common in smaller vessels. Blood (containing hemoglobin) or other physiological fluids containing ionic constituents in the human body respond to magnetic body forces when subjected to external (extracorporeal) magnetic fields. Inspired by such applications, in the present work we have considered the forced convective flow of an electrically conducting viscoelastic physiological fluid through a ciliated channel under the action of a transverse magnetic field. The presence of deposits (fats, cholesterol, etc.) in the channel is mimicked with a Darcy porous medium drag force model. The effect of energy loss is simulated via the inclusion of viscous dissipation in the energy conservation (heat) equation. The velocity, temperature, and pressure distribution are computed in the form of infinite series constructed by Adomian decomposition method and numerically evaluated in a symbolic software (Mathematica). The influence of Hartmann number (magnetic parameter), Jeffrey first and second viscoelastic parameters, permeability parameter (modified Darcy number), and Brinkman number (viscous heating parameter) on velocity, temperature, pressure gradient, and bolus dynamics is visualized graphically.

Theory of Scalar Wave Scattering by a Sphere and a Planar Substrate

The problem of scalar wave scattering by a sphere on or near a planar substrate is analytically solved. The solution is a set of wave functions coming in the form of infinite series of spherical and plane waves. In air, the incident plane wave is either scattered by the sphere or reflected from the substrate. A part of these scattered or reflected waves propagate to the other object where it is reflected and scattered again. Such processes of scattering and reflection repeat in turn indefinitely to generate multiply scattered waves, which are represented in the corresponding terms in the infinite series. The term in the series can be arranged in a recognizable manner to explicitly reveal the involved process and the multiplicity of scattering.

Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator

In this paper, we consider some linear/nonlinear differential equations (DEs) containing conformable derivative operator. We obtain approximate solutions of these mentioned DEs in the form of infinite series which converges rapidly to their exact values by using and homotopy analysis method (HAM) and modified homotopy perturbation method (MHPM). Using the conformable operator in solutions of different types of DEs makes the solution steps are computable easily. Especially, the conformable operator has been used in modelling DEs and identifying particular problems such as biological, engineering, economic sciences and other some important fields of application. In this context, the aim of this study is to solve some illustrative linear/nonlinear problems as mathematically and to compare the exact solutions with the obtained solutions by considering some plots. Moreover, it is an aim to show the authenticity, applicability, and suitability of the methods constructed with the conformable operator.

Volterra bound interval type‐2 fuzzy logic‐based approach for multiple power quality events analysis

This study demonstrates detection and classification of power quality (PQ) events utilising Volterra series for feature extraction and interval type-2 fuzzy logic system (IT2FLS) for classification of PQ events. The Volterra series represented in the form of infinite power series with memory which provides a convenient and strong platform for representation of input–output relationship for non-linear systems. IT2FLS uses the concept of membership functions to perform classification of multiple PQ events. When supply power is distorted by additive noise where signal-to-noise ratio is low and uncertain, IT2FLS has shown improved performance over support vector machine, neural networks (NNs), probabilistic NN and type-1 fuzzy logic system classifiers, which makes an IT2FLS favourable for real-time applications.

The influence of boundaries on sound insulation of the multilayered aerospace poroelastic composite structure

Acoustic analysis of the four-sides simply supported doubly curved composite shell interlayered with porous material used in aerospace applications is considered based on Third order Shear Deformation Theory (TSDT). The focus is specifically placed on presenting the effect of boundaries on Sound Transmission Loss (STL) of the poroelastic structure. Then, the results are compared with those of infinite shell. In order to calculate the STL, the displacement and pressure terms are derived in the form of double Fourier series for a simply supported boundary condition doubly curved shell. A new approach is made to consider the number of modes for the finite composite structure treated with porous material. In addition, the simultaneous solution of the equations of porous and composite layers along with acoustic wave equations is performed. This process leads to obtain the unknown constants in all parts of the structure including upper and bottom composite shells as well as porous core which results in presenting the STL of the structure in the logarithmic scale. Since, the solution is provided in the form of modal infinite series which should be terminated at the sufficient modes, the convergence checking is prepared in 3D configurations with respect to various frequencies and shell dimensions to present the number of the appropriate modes for this process. Moreover, the results are validated with the aid of other researches including experimental as well as analytical models available in literature. Finally, some new results including Sound Pressure Level (SPL) in both of global view and contour map, transverse displacement configurations and the effects of various shell dimensions on STL, are presented for this finite structure.

English

This paper examines a general recurrence relation by the use of fractional reduced differential transform and then a scheme (methodology) on how to find closed solutions of one dimensional time fractional diffusion equations with initial conditions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions by the use of fractional reduced differential transform method. The new general recurrence relation and the methodology of the fractional reduced differential transform method were successfully developed. The obtained new general recurrence relation helps us to solve time-fractional diffusion equations with initial conditions and various external forces by using fractional reduced differential transform method. To see its effectiveness and applicability, five test examples were presented. The results show that the general recurrence relation works successfully in solving time-fractional diffusion equations in a direct way without using linearization, transformations, perturbation, discretization or restrictive assumptions by using fractional reduced differential transform method.   Key words: Time fractional diffusion equations with initial conditions, Caputo fractional derivatives, Mittag-Leffler function, Fractional reduced differential transform method (FRDTM).

Thermoelastic damping in micromechanical resonators operating as mass sensors

This paper presents an analytical study on evaluation of thermoelastic damping (TED) in micromechanical resonators operating as mass sensors by means of the thermal energy approach. Both quasi-one-dimensional model and two-dimensional model are proposed and analytical expressions of TED in infinite series form for micro-beam resonator-based mass sensors are derived. We apply the proposed models to examine effects of various factors on TED, such as the position of attached mass particle, boundary conditions, geometry of resonators and mode shape of vibration. Numerical results show that thermoelastic damping of micro-beam resonator-based mass sensors increases obviously with the increase of the mass ratio of the attached mass to mass of the resonator. Compared with results of TED obtained by 2-D model accounting for the two-dimensional thermal conduction in micro-beam resonators, the quasi-1-D model overestimates TED in the high frequency range, and underestimates TED in the low frequency range. Noticeable differences are observed in the results of TED between quasi-1-D model and 2-D model. We also obtain the expressions of mass sensitivity and minimum detectable mass imposed by thermomechanical noise process for resonant mass sensors of bridge and cantilever configurations.

Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks

In this paper, a rumor diffusion model with uncertainty of human behavior under spatio-temporal diffusion framework is established. Take physical significance of spatial diffusion into account, a diffusion threshold is set under which the rumor is not a trend topic and only spreads along determined physical connections. Heterogeneity of degree distribution and distance distribution has also been considered in theoretical model at the same time. The global existence and uniqueness of classical solution are proved with a Lyapunov function and an approximate classical solution in form of infinite series is constructed with a system of eigenfunction. Simulations and numerical solutions both on Watts–Strogatz (WS) network and Barabási–Albert (BA) network display the variation of density of infected connections from spatial and temporal dimensions. Relevant results show that the density of infected connections is dominated by network topology and uncertainty of human behavior at threshold time. With increase of social capability, rumor diffuses to the steady state in a higher speed. And the variation trends of diffusion size with uncertainty are diverse on different artificial networks.

New Stability Criteria of Delayed Load Frequency Control Systems via Infinite-Series-Based Inequality

A new approach is proposed for the stability problem of delayed load frequency control (LFC) scheme with fixed and time-varying delay cases included in the current paper. New stability criteria with delay dependence in terms of linear matrix inequalities for LFC systems are derived by a novel augmented Lyapunov–Krasovski (L–K) functional. Our proof deployment for system stability of power grids employs the further improved integral inequality in the form of infinite series, which turns out to be less conservative than Wirtinger's inequality that encompasses Jensen inequality. Simulation case studies are carried out to show the effectiveness and superiority of the presented delay-dependent PI-type LFC design scheme.

Ruin probabilities in classical risk models with gamma claims

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.

Thermoelastic Damping in Micro- and Nanobeam Resonators With Non-Fourier Heat Conduction

Predicting thermoelastic damping (TED) under the effect of non-Fourier heat conduction is challenging and crucial for the design of micro- and nanobeam resonators with high quality factors. In the past, many works were devoted to the modeling of TED under the effect of non-Fourier heat conduction. Some analytical and numerical results for TED were presented. However, the spectrum of TED contains multiple peaks only in the numerical results. The previous analytical model failed to give a reasonable prediction for the multiple-peak phenomenon. Few works can give a convincing explanation for the existence of multiple peaks. In this paper, the TED in micro- and nanobeam resonators under the effect of non-Fourier heat conduction is investigated using the generalized thermoelasticity theory with single-phase-lag time. An analytical formula of TED is obtained in the form of infinite series. The effects of non-Fourier heat conduction on TED are studied for representative cases of the micro- and nanobeams. A simple analytical expression for the accurate determination of the multiple peaks in TED is obtained. The results of the present model agree well with the numerical results available in the reported literature. The results of this paper may bring new insights for the multiple-peak phenomenon of TED spectrum in micro- and nanobeams.

Approximate solution fuzzy pantograph equation by using homotopy perturbation method

In this paper, Homotopy Perturbation Method (HPM) is modified and formulated to find the approximate solution for its employment to solve (FDDEs) involving a fuzzy pantograph equation. The solution that can be obtained by using HPM is in the form of infinite series that converge to the actual solution of the FDDE and this is one of the benefits of this method In addition, it can be used for solving high order fuzzy delay differential equations directly without reduction to a first order system. Moreover, the accuracy of HPM can be detected without needing the exact solution. The HPM is studied for fuzzy initial value problems involving pantograph equation. Using the properties of fuzzy set theory, we reformulate the standard approximate method of HPM and obtain the approximate solutions. The effectiveness of the proposed method is demonstrated for third order fuzzy pantograph equation.

On one boundary-value problem for an equation of higher even order

We study boundary-value problem for an equation of even order in a rectangular domain. By the spectralmethod we obtain necessary and sufficient conditions of uniqueness of a solution. The solution is constructed in the formof infinite series in eigenfunctions. We obtain sufficient conditions under which this series is a regular solution.

Energy Detection in Multi-user Relay Networks

In this correspondence, detection performance of an energy detector is evaluated in multi-user relay networks equipped with single amplify-and-forward (AF) relay. The relay network is assumed to be operating over independent and identically distributed, flat Rayleigh fading channels. New expressions for average detection probability are derived for both fixed-gain and variable-gain AF relays. Furthermore, truncation error bounds are also evaluated for the respective expressions because the derived average detection probability expressions were in the form of infinite summation series. In the end, the accuracy of the derived expressions is validated with the help of simulations .

Distribution of absolute maximum of mean square differentiable Gaussian stationery process

In this paper it is obtained the distribution of absolute maximum of mean square differentiable stationery Gaussian process by means of integration of the results of the second Kolmogorov equation solution. It is shown the way simplifying integration and its interrelation to integro-differential equation obtained before. The second Kolmogorov equation is solved first for the boundary conditions allowing to obtain the results in form of infinite series with coefficients obtained by means of solution of Sturm–Liouville problem and reducing to the simple expression. It is analyzed the correlation of obtained results with known before. It is carried out a comparative analysis of correlation functions and expressions for distribution of absolute maximums of mean square differentiable and single-component Markov processes. In spite of correlation function of single-component Markov process can be considered as limit expression for correlation function of mean square differentiable process, the expression for distribution of their absolute maximums are essentially different. It shows practical meaning of the results since real processes in radio engineering systems can be mean square differentiable only.